Active Noise Control For Periodic Disturbances - American Control ...
Proceedings of the American Control Conference Philadelphia, Pennsylvania June 1998
Active Noise Control for Periodic Disturbances Marc Bodson, Jonathan S. Jensen and Scott C. Douglas * Department of Electrical Engineering, University of Utah Salt Lake City, UT 84112, U.S.A. Abstract: This paper proposes an active noise control algorithm for periodic disturbances of unknown frequency. The algorithm is appropriate for the feedback case in which a single error microphone is used. A previously-proposed algorithm for the rejection of sinusoidal noise sources is extended for the cancellation of multiple harmonics. Unlike many other approaches, the estimates of the frequencies of the separate harmonics are tied together within the algorithm to account for the integer multiplicative relations between them. The dynamic behavior of the closed-loop system is analyzed using an approximation that is shown, in simulations, to provide an accurate representation of the system's behavior. Experimental results on an active noise control testbed demonstrate the success of the method in a practical environment.
This paper extends the algorithm of [6] to the more general case of periodic disturbances with multiple harmonics. A particular feature of the new algorithm is its use of the integer relationships between the harmonic components of the disturbance within the adaptive algorithm.
1. Introduction
where y(s), u(s) and d ( s ) are the Laplace transforms of the microphone signal, of the speaker output, and of the equivalent noise signal at the speaker location, respectively. Alternatively, u ( t ) may be viewed as the control input, d ( t ) as the disturbance, and y(t) as the plant output. The goal of the control system is to generate u ( t ) such that y(t) + 0 as t + CO. The objective would be achieved if u ( t ) = d ( t ) , but the disturbance d ( t ) is not known or measured in any way except through its effect at the output of the system. It is assumed t o be a periodic signal, so that
The problem of active noise control is considered, as shown in Fig. 1. A microphone is used to measure the instantaneous noise level at some location to be made quiet. The signal is sampled and then processed by a digital signal processing system, and an anti-noise field is generated through a loudspeaker. The objective is to eliminate or significantly reduce the noise level at the microphone through destructive interference.
2. Adaptive Algorithm 2.1 Problem Statement Assume that the transformation from the speaker to the microphone is a stable linear time-invariant system with transfer function P ( s ) and that the effect of the noise source is additive. In the Laplace domain, the system can be modelled as Y(S)
= P ( s ) ( u ( s )- d ( s ) ) ,
(1)
n
d(t)
=
x
d
k Cos(ak,d(t)),
k=l &k,d(t)
Figure 1: Active noise control (feedback scheme) As shown in Fig. 1, the system configuration under consideration uses only one microphone. In other words, the situation is of a pure feedback nature, in contrast to the feedforward set-up that is often considered in such applications [l].Moreover, the noise is assumed to be periodic in nature so that, from a control perspective, the problem is a classical rejection problem for periodic disturbances, except that the frequency of the disturbance is unknown and potentially time-varying. A limited number of approaches exist to address this problem. They include adaptive algorithms employing the internal model principle [2, 3, 41 and extensions of adaptive algorithms for disturbances of known frequency [5, 61.
=
kW1.
(2)
The parameters w1, d k , and a k , d ( O ) (for k = 1, ..., n ) are unknown. The order of the highest harmonic, n , is assumed to be finite and known. Certain harmonics may also be specified to be absent, z.e., certain values of d k may be known to be small a priori. For simplicity of presentation, we will consider the case where the fundamental and the third harmonic are present (only d l and d3 are nonzero).
2.2 Adaptive Algorithm
The structure of the proposed scheme is shown in Fig. 2. The signal u 1 nominally cancels the fundamental (at frequency w I ) , while the signal u3 cancels the third harmonic. The parameters a1 and a3 are the estimates of the angles of the two components of the disturbance. The parameters 011 and 6'31 are the estimates of the magnitudes of the two sinusoids, and 0 1 2 is the estimate of the frequency of the fundamental. The value of 812 is inte*This material is based upon work supported by the U S . Army grated to obtain the angle a l . For the third harmonic, Research Office under grant number DAAH04-96-1-0085. The con0 3 2 is not the frequency but rather is the relative phase of tent of the paper does not necessarily reflect the position or the polthe signal. The algorithm uses the assumption that the icy of the federal government, and no official endorsement should be inferred. second sinusoid is a third harmonic of the fundamental 2616 0-7803-4530-4/98 $10.000 1998 AACC
Authorized licensed use limited to: SOUTHERN METHODIST UNIV. Downloaded on October 31, 2008 at 17:34 from IEEE Xplore. Restrictions apply.
Y
U
The update laws are defined through integral relationships to guarantee zero steady-state errors. The control law for 6'12 is slightly different from the others, with the signal 2 1 2 filtered so that
>
s+a
212f(s)
I
!
I
I
I
(8)
s+b
The compensation filter is necessary to ensure the stability of the closed-loop system. The constants a, b, 9 1 , g 2 , and g 3 will be adjusted to obtain satisfactory performance. The parameters' dl, and d3, are estimates of dl and d3 (see section 3 . 2 ) . To extend the algorithm for arbitrary harmonics, additional paths similar to that for the third harmonic in Fig. 2 may be added. The multiplyingfactor of 3, the matrix G 3 , and the estimate d~~are the only elements that need to be changed. Note that, in the proposed implementation, frequency estimation is provided through the first harmonic. This choice is not essential, and frequency estimation based on another harmonic is possible.
e3 1 I
= F(s)zIz(s), F ( s ) = -.
-sin(ag)
Figure 2: Adaptive Algorithm
3. Stability Analysis and Control Design by letting the angle a 3 be equal to the sum of three times the angle of the fundamental and of the relative phase 632.
The equations for the control algorithm are thus
+
U
=
6'11 C O S ( ( Y ~631 ) COS(Q~)
kl
=
o12,
Y11
931
a3
= 3 ' a 1+e32
= Y cos(a1), Y12 = -Y sin(a1) = Y C O S ( Q ~ )Y32 , = -Y sin(a3).
(3)
The parameters have nominal values OTl = d l , = w1, e;, = d 3 , and 6'& = a 3 , d ( 0 ) - 3 a l , d ( 0 ) . For these values, and for crl(t) - w l t = a l , d ( O ) , the output converges to zero. The transfer function matrix Cl(s) relates the signals y11 and y12 to the parameters 611 and 0 1 2 , respectively. Similarly, the transfer function matrix C 3 ( s ) relates the signals y31 and 5132 to the parameters 6'31 and 6'32. The ) the products of constant matrices Cl(s) and C ~ ( Sare matrices with diagonal transfer function matrices. They are defined as follows. Consider the real and imaginary parts of the plant frequency response at the two frequencies of interest, given by pR,1
pI,l
= Re[P(@l)], = Im[P(jwl)],
pR,3 pI,3
= Re[P(3jul)]i = Im[P(3jwl)]i
(4)
3.1 Stability Analysis Our analysis of the system is based on a fundamental fact, whose proof is reminiscent of derivations found in the study of frequency-modulation communication systems [7] and of averaging methods applied to adaptive systems [$I. The following conditions are assumed: 0 the values of 0 1 1 , 6'12, 6'31, and 6'32 vary sufficiently slowly that the reslponse of the plant to the signal u(t) may be approximated by the steady-state output of the plant for the two sinusoidal signals with frequencies 612 and 36'12. a the instantaneous frequency 612 is close to w 1 , so that P(j6'2) may be replaced by P ( j w 1 ) and p(3j&)may be replaced by P ( 3 j w l ) . Basic Fact: Considering low-frequency components only, the signals q ~ ( t ) q, ~ ( t ) 2,3 1 ( t ) , and 2 3 2 ( t ) are approximately given by
W(t) ( %(t) ) = 21 (
211, 212, 231,
al(t) - a l , d ( t ) = a3(t)
and
232
are defined as
and the algorithm parameters are given by 411 031
=
-291211,
61,
= -2g2212f/dle,
-293231,
d32
= -2g3232/d3ea
)
(9)
for i = {1,3}, with
and define the matrices
Variables
&(t) - er1 cos(ai(t) - Q i , d ( t ) ) Orl sin(aj(t) - ai,d(t))
1'
(b!(U)
+
- @:2)dC m l ( 0 )
-al,d(O)i
- a 3 , d ( t ) = 3(ai(t) - W , d ( t ) ) -k 6 3 2 ( t ) -6;~.
(10)
The proof follows similar steps as in the proof in [6] and is omitted. The elimination of the high-frequency components within the system (can be achieved through low-pass filtering of the signals. However, the signals are filtered within the compensators Cl(s) and C3(s), and although the filtering is not ideal, simulations show that it is sufficient for the satisfactory operation of the system. 3.2 Compensator Design Although the equations are nonlinear, a linear system is obtained if the parameters are close to their nominal
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values and the phase error S a l ( t ) = small. The linearized system is
al(t)
The linearized dynamics from the parameters 011 and 812 to the variables 2 1 1 and 2 1 2 are decoupled from one another and are not dependent on the dynamics of the variables associated with the third harmonic. In closed-loop, the dynamics of 811 are those of a first-order system with a pole at s = -91. For 0 1 2 , the closed-loop poles are determined by the roots of s2(s b ) g2(s a ) = 0, if d l , = d l = 07,. Otherwise, g2 is replaced by g 2 d l / d l e . Stability is guaranteed if g2 > 0 and b > a > 0. For the variables associated to the third harmonic, the closedloop dynamics are those of a first-order system with a pole at s = -93. For 832, the pole is at s = -g3d3/d3e if is not equal to d3 = S;, . The phase error Sal(t) also appears as a disturbance on the equation for 6 3 2 . The stability of the equation for 0 1 2 ensures that this disturbance vanishes with time. Because the magnitudes of the sinusoidal components d l and d3 act as gains in the two transfer functions associated with phase locking, estimates of the parameters are used to ensure that the closed-loop poles are set at desirable values. However, the stability of the linear systems is not dependent upon the accuracy of these estimates. The algorithm also requires the knowledge of the matrices G1 and G3, which depend on w 1 . One may choose to set these matrices for a value of the frequency in the middle of the expected range of operation, or one may use the estimated frequency 012 in real-time. Either way, knowledge of the frequency response of the plant in the frequency range of interest is required.
+ +
1.02,
- a l , d ( t ) is
0
+
I
0.1
I
0.2
,
,
0.3 0.4
,
,
I
0.5 0.6 0.7
,
,
,
0.8
0.9
1
Time (seconds)
Figure 4: Magnitude of the fundamental 106,
0
I
0.1
(811)
,
0.2 0.3 0.4
0.5
0.6
0.7
0.8 0.9
I
Time (seconds)
Figure 5: Frequency of the fundamental
(012)
and 93 are set to 10, leading to closed-loop poles for the first-order systems at -10 rad/s. The other parame4. Simulation Results ters are set to 92 = 400, a = 5 , and b = 30, leading to closed-loop poles for the frequency control loop located at 0.8 -10 rad/s and -10 fj l 0 rad/s. The initial states of the parameters are zero, except for 811(0) = 0.9, e1,(0) = 90 0.6 rad/s. Fig. 3 shows the output of the plant, which is found to 0.4 decrease to negligible values in less than a second. The 5 0.2 transient behavior of the magnitude estimate 811 is shown a 3 in Fig. 4, where the solid line is the parameter response, g o and the dashed line is the response predicted from a simm ulation of an approximate system. The approximate sys-0.2 tem is composed of the control law (7), ( S ) , and the nonlinear approximation (9), (10). The frequency estimate -0.r 812 is shown in Fig. 5. The matches between the actual -0.f and approximate behaviors is good, and the degree of match is similar to those for other adaptive systems [SI. -0l The magnitude estimate for the third harmonic 831 is 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (seconds) shown in Fig. 6 and the phase estimate 832 in Fig. 7. Figure 3: Plant output (y) Note that the magnitude e;, was set to -1, and the estiThe performance of the algorithm is examined via simumate of the magnitude converged to 1, with the estimate lation. We consider a situation in which P ( s ) = lOO/(s+ of the phase converging to - T , or -180'. Again, the approximations are very good, and although the nonlinl o o ) , W I = 100, d l = 1, and d3 = -1. The parameters 2618 gl
I
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ear effects are significant, the design based on the linear approximation is adequate to obtain convergence of the system
-
t
-0.06
-0.08 0
0.5
I I
1.5
Time (seconds)
Figure 8: Microphone signal -0.41
0
"
0.1
"
"
0.2 0.3 0.4 0.5 0.6 Time (seconds)
"
0.7
0.8
'
0.9
I
1
Figure 6: Magnitude of the third harmonic (Ow 1
1 0-5 .-
I 0
0.5
1
1.5
Time (seconds)
Figure 9: Magnitudse of the fundamental 0.8
0.9
Figure 7: Phase of the third harmonic
(032)
0
0.1
0.5 0.6 Time (seconds)
0.2 0.3 0.4
0.7
(011)
1
were obtained at 16 different frequencies, spaced logarithmically between 32.5 Hz a.nd 1kHz. In other experiments, 5. Experimental Results the frequency was calculated from a 50-tap finite - response _ impulse response model obtained with an adaptive idenThe scheme was implemented on an experimental actification algorithm and at white noise input. This identitive noise control system developed a t the University of fication procedure took longer. The matrices G1 and G3 Utah. The algorithm was coded in assembly language on were adjusted in real-time, based on the frequency estia Motorola DSP96002 32-bit floating-point digital signal mate 012. Values from a look-up table were interpolated processorl The sampling rate was set at 8 kHz. A sinlinearly as needed. UPdatse ofthe matrices was Performed gle bookshelf speaker with a 4-inch low-frequency driver, every 8 samples. Because of the digital implementation, located approximately 2 ft away from the error microa discrete-time equivalent of the algorithm was implephone, generated a periodic signal constituting the noise mented, with the z-domain poles placed in the vicinity source. The microphone signal was passed through an of z = 0.995 for the para.meters of the fundamental and anti-aliasing filter and sampled by a self-calibrating 16z = 0.99 for the parameters of the third harmonic. bit analog-to-digital converter before being sent to the The results of one experiment are shown on Fig. 8, in DSP system. The controller output signal was sent to a which the signal at the error microphone is plotted as a noise cancelling speaker placed approximately 1 ft away function of time. The algorithm is not engaged until 0.5 from the microphone. Only a single error sensing microsec. in the experiment, so that the amount of noise before phone signal was used. compensation can be judged. The frequency of the fundaThe frequency response of the plant was determined mental is 110 Hz. The plot shows that the algorithm, once during a rapid calibration phase in which pure sinusoidal engaged, cancels the noise within a fraction of a second. tones were applied to the noise cancellation speaker, and The adaptive parameters for the fundamental are shown the responses were measured by the microphone signal. in Fig. 9 (magnitude estimate 811) and Fig. 10 (frequency The real and imaginary parts of the frequency response 2619
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0.12 I
0
I
0.5
1
I._
I .5
0
Time (seconds)
0.05
3
I
-
,
Figure 12: Phase of the third harmonic
I
-0.05E =
(U
s.-
-0.1
-
-0.15
-
7. References
C
5
(032)
nonlinear time-varying system. However, we showed that an approximate nonlinear time-invariant system provided a very accurate representation of the dynamic behavior of the system. A further approximation of the system through linearization was useful for the selection of the design parameters. The algorithm was tested experimentally on an active noise control system at the University of Utah to demonstrate the success of the method in a practical environment.
1
0
W
1.5
Time (seconds)
Figure 10: Frequency of the fundamental (012) 0.1
1
0.5
S.M. Kuo and D.R. Morgan, Active Noise Control Systems: Algorilhms and DSP Implementations, New York, Wiley, 1996.
”.*
0
0.5
1
1.5
Time (seconds)
K.S. Narendra and A. Annaswamy, Stable Adaptive
Figure 11: Magnitude of the third harmonic (03,)
Systems, Prentice-Hall, Englewood Cliffs, NJ, 1989.
estimate 012). The units of the frequency estimate are given in radians/sample; L e , 27r . 110/8000 = 0.086. The initial frequency estimate corresponded to 115 He. Other parameters were initially set to zero. Shown in Fig. 11 and 12 are the parameters related to the third harmonic (magnitude estimate 031 and phase estimate 032 (in rad)). It was found that the frequency of the disturbance could vary over a wide range, once the algorithm had locked onto the disturbance frequency. Techniques from phaselocked loops could be used to expand the lock-in range [7] but were not implemented. Other experiments employing several harmonics as well as frequency estimation using high-order harmonics provided results comparable to those shown.
G. Feng and M. Palaniswamy, “Unified Treatment of Internal Model Principle Based Adaptive Control Algorithm,” Int. J. Control, vol. 54, no. 4, pp. 883-901, 1991. G. Feng and M . Palaniswamy, “A Stable Adaptive Implementation of the Internal Model Principle,” IEEE Trans. on Automatic Control, vol. 37, no. 8, pp. 12201225, 1992. G.B.B. Chaplin and R.A. Smith, “Method of and Apparatus for Cancelling Vibrations from a Source of Repetitive Vibrations,” U. S. Patent 4,566,118, Jan. 21, 1986. M. Bodson and S.C. Douglas, “Adaptive Algorithms for the Rejection of Periodic Disturbances with Un-
6. Conclusions
known Frequency,” Automatica, vol. 33, no. 12, pp. 2213-2221, 1997.
An adaptive algorithm is proposed for the rejection of periodic disturbances of unknown frequency. For simplicity, the algorithm is described for a noise consisting of a fundamental component and a third harmonic, although the method is easily extended to noises with an arbitrary number of harmonics. As in other solutions to this control problem, the closed-loop system is a complex
J.R. Smith, Modern Communzcations Circuits, Mc Graw-Hill, New York, NY, 1997.
S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness, Prentice-Hall, Englewood Cliffs, NJ, 1989.
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Active Noise Control for Periodic Disturbances Marc Bodson, Jonathan S. Jensen and Scott C. Douglas * Department of Electrical Engineering, University of Utah Salt Lake City, UT 84112, U.S.A. Abstract: This paper proposes an active noise control algorithm for periodic disturbances of unknown frequency. The algorithm is appropriate for the feedback case in which a single error microphone is used. A previously-proposed algorithm for the rejection of sinusoidal noise sources is extended for the cancellation of multiple harmonics. Unlike many other approaches, the estimates of the frequencies of the separate harmonics are tied together within the algorithm to account for the integer multiplicative relations between them. The dynamic behavior of the closed-loop system is analyzed using an approximation that is shown, in simulations, to provide an accurate representation of the system's behavior. Experimental results on an active noise control testbed demonstrate the success of the method in a practical environment.
This paper extends the algorithm of [6] to the more general case of periodic disturbances with multiple harmonics. A particular feature of the new algorithm is its use of the integer relationships between the harmonic components of the disturbance within the adaptive algorithm.
1. Introduction
where y(s), u(s) and d ( s ) are the Laplace transforms of the microphone signal, of the speaker output, and of the equivalent noise signal at the speaker location, respectively. Alternatively, u ( t ) may be viewed as the control input, d ( t ) as the disturbance, and y(t) as the plant output. The goal of the control system is to generate u ( t ) such that y(t) + 0 as t + CO. The objective would be achieved if u ( t ) = d ( t ) , but the disturbance d ( t ) is not known or measured in any way except through its effect at the output of the system. It is assumed t o be a periodic signal, so that
The problem of active noise control is considered, as shown in Fig. 1. A microphone is used to measure the instantaneous noise level at some location to be made quiet. The signal is sampled and then processed by a digital signal processing system, and an anti-noise field is generated through a loudspeaker. The objective is to eliminate or significantly reduce the noise level at the microphone through destructive interference.
2. Adaptive Algorithm 2.1 Problem Statement Assume that the transformation from the speaker to the microphone is a stable linear time-invariant system with transfer function P ( s ) and that the effect of the noise source is additive. In the Laplace domain, the system can be modelled as Y(S)
= P ( s ) ( u ( s )- d ( s ) ) ,
(1)
n
d(t)
=
x
d
k Cos(ak,d(t)),
k=l &k,d(t)
Figure 1: Active noise control (feedback scheme) As shown in Fig. 1, the system configuration under consideration uses only one microphone. In other words, the situation is of a pure feedback nature, in contrast to the feedforward set-up that is often considered in such applications [l].Moreover, the noise is assumed to be periodic in nature so that, from a control perspective, the problem is a classical rejection problem for periodic disturbances, except that the frequency of the disturbance is unknown and potentially time-varying. A limited number of approaches exist to address this problem. They include adaptive algorithms employing the internal model principle [2, 3, 41 and extensions of adaptive algorithms for disturbances of known frequency [5, 61.
=
kW1.
(2)
The parameters w1, d k , and a k , d ( O ) (for k = 1, ..., n ) are unknown. The order of the highest harmonic, n , is assumed to be finite and known. Certain harmonics may also be specified to be absent, z.e., certain values of d k may be known to be small a priori. For simplicity of presentation, we will consider the case where the fundamental and the third harmonic are present (only d l and d3 are nonzero).
2.2 Adaptive Algorithm
The structure of the proposed scheme is shown in Fig. 2. The signal u 1 nominally cancels the fundamental (at frequency w I ) , while the signal u3 cancels the third harmonic. The parameters a1 and a3 are the estimates of the angles of the two components of the disturbance. The parameters 011 and 6'31 are the estimates of the magnitudes of the two sinusoids, and 0 1 2 is the estimate of the frequency of the fundamental. The value of 812 is inte*This material is based upon work supported by the U S . Army grated to obtain the angle a l . For the third harmonic, Research Office under grant number DAAH04-96-1-0085. The con0 3 2 is not the frequency but rather is the relative phase of tent of the paper does not necessarily reflect the position or the polthe signal. The algorithm uses the assumption that the icy of the federal government, and no official endorsement should be inferred. second sinusoid is a third harmonic of the fundamental 2616 0-7803-4530-4/98 $10.000 1998 AACC
Authorized licensed use limited to: SOUTHERN METHODIST UNIV. Downloaded on October 31, 2008 at 17:34 from IEEE Xplore. Restrictions apply.
Y
U
The update laws are defined through integral relationships to guarantee zero steady-state errors. The control law for 6'12 is slightly different from the others, with the signal 2 1 2 filtered so that
>
s+a
212f(s)
I
!
I
I
I
(8)
s+b
The compensation filter is necessary to ensure the stability of the closed-loop system. The constants a, b, 9 1 , g 2 , and g 3 will be adjusted to obtain satisfactory performance. The parameters' dl, and d3, are estimates of dl and d3 (see section 3 . 2 ) . To extend the algorithm for arbitrary harmonics, additional paths similar to that for the third harmonic in Fig. 2 may be added. The multiplyingfactor of 3, the matrix G 3 , and the estimate d~~are the only elements that need to be changed. Note that, in the proposed implementation, frequency estimation is provided through the first harmonic. This choice is not essential, and frequency estimation based on another harmonic is possible.
e3 1 I
= F(s)zIz(s), F ( s ) = -.
-sin(ag)
Figure 2: Adaptive Algorithm
3. Stability Analysis and Control Design by letting the angle a 3 be equal to the sum of three times the angle of the fundamental and of the relative phase 632.
The equations for the control algorithm are thus
+
U
=
6'11 C O S ( ( Y ~631 ) COS(Q~)
kl
=
o12,
Y11
931
a3
= 3 ' a 1+e32
= Y cos(a1), Y12 = -Y sin(a1) = Y C O S ( Q ~ )Y32 , = -Y sin(a3).
(3)
The parameters have nominal values OTl = d l , = w1, e;, = d 3 , and 6'& = a 3 , d ( 0 ) - 3 a l , d ( 0 ) . For these values, and for crl(t) - w l t = a l , d ( O ) , the output converges to zero. The transfer function matrix Cl(s) relates the signals y11 and y12 to the parameters 611 and 0 1 2 , respectively. Similarly, the transfer function matrix C 3 ( s ) relates the signals y31 and 5132 to the parameters 6'31 and 6'32. The ) the products of constant matrices Cl(s) and C ~ ( Sare matrices with diagonal transfer function matrices. They are defined as follows. Consider the real and imaginary parts of the plant frequency response at the two frequencies of interest, given by pR,1
pI,l
= Re[P(@l)], = Im[P(jwl)],
pR,3 pI,3
= Re[P(3jul)]i = Im[P(3jwl)]i
(4)
3.1 Stability Analysis Our analysis of the system is based on a fundamental fact, whose proof is reminiscent of derivations found in the study of frequency-modulation communication systems [7] and of averaging methods applied to adaptive systems [$I. The following conditions are assumed: 0 the values of 0 1 1 , 6'12, 6'31, and 6'32 vary sufficiently slowly that the reslponse of the plant to the signal u(t) may be approximated by the steady-state output of the plant for the two sinusoidal signals with frequencies 612 and 36'12. a the instantaneous frequency 612 is close to w 1 , so that P(j6'2) may be replaced by P ( j w 1 ) and p(3j&)may be replaced by P ( 3 j w l ) . Basic Fact: Considering low-frequency components only, the signals q ~ ( t ) q, ~ ( t ) 2,3 1 ( t ) , and 2 3 2 ( t ) are approximately given by
W(t) ( %(t) ) = 21 (
211, 212, 231,
al(t) - a l , d ( t ) = a3(t)
and
232
are defined as
and the algorithm parameters are given by 411 031
=
-291211,
61,
= -2g2212f/dle,
-293231,
d32
= -2g3232/d3ea
)
(9)
for i = {1,3}, with
and define the matrices
Variables
&(t) - er1 cos(ai(t) - Q i , d ( t ) ) Orl sin(aj(t) - ai,d(t))
1'
(b!(U)
+
- @:2)dC m l ( 0 )
-al,d(O)i
- a 3 , d ( t ) = 3(ai(t) - W , d ( t ) ) -k 6 3 2 ( t ) -6;~.
(10)
The proof follows similar steps as in the proof in [6] and is omitted. The elimination of the high-frequency components within the system (can be achieved through low-pass filtering of the signals. However, the signals are filtered within the compensators Cl(s) and C3(s), and although the filtering is not ideal, simulations show that it is sufficient for the satisfactory operation of the system. 3.2 Compensator Design Although the equations are nonlinear, a linear system is obtained if the parameters are close to their nominal
2617
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values and the phase error S a l ( t ) = small. The linearized system is
al(t)
The linearized dynamics from the parameters 011 and 812 to the variables 2 1 1 and 2 1 2 are decoupled from one another and are not dependent on the dynamics of the variables associated with the third harmonic. In closed-loop, the dynamics of 811 are those of a first-order system with a pole at s = -91. For 0 1 2 , the closed-loop poles are determined by the roots of s2(s b ) g2(s a ) = 0, if d l , = d l = 07,. Otherwise, g2 is replaced by g 2 d l / d l e . Stability is guaranteed if g2 > 0 and b > a > 0. For the variables associated to the third harmonic, the closedloop dynamics are those of a first-order system with a pole at s = -93. For 832, the pole is at s = -g3d3/d3e if is not equal to d3 = S;, . The phase error Sal(t) also appears as a disturbance on the equation for 6 3 2 . The stability of the equation for 0 1 2 ensures that this disturbance vanishes with time. Because the magnitudes of the sinusoidal components d l and d3 act as gains in the two transfer functions associated with phase locking, estimates of the parameters are used to ensure that the closed-loop poles are set at desirable values. However, the stability of the linear systems is not dependent upon the accuracy of these estimates. The algorithm also requires the knowledge of the matrices G1 and G3, which depend on w 1 . One may choose to set these matrices for a value of the frequency in the middle of the expected range of operation, or one may use the estimated frequency 012 in real-time. Either way, knowledge of the frequency response of the plant in the frequency range of interest is required.
+ +
1.02,
- a l , d ( t ) is
0
+
I
0.1
I
0.2
,
,
0.3 0.4
,
,
I
0.5 0.6 0.7
,
,
,
0.8
0.9
1
Time (seconds)
Figure 4: Magnitude of the fundamental 106,
0
I
0.1
(811)
,
0.2 0.3 0.4
0.5
0.6
0.7
0.8 0.9
I
Time (seconds)
Figure 5: Frequency of the fundamental
(012)
and 93 are set to 10, leading to closed-loop poles for the first-order systems at -10 rad/s. The other parame4. Simulation Results ters are set to 92 = 400, a = 5 , and b = 30, leading to closed-loop poles for the frequency control loop located at 0.8 -10 rad/s and -10 fj l 0 rad/s. The initial states of the parameters are zero, except for 811(0) = 0.9, e1,(0) = 90 0.6 rad/s. Fig. 3 shows the output of the plant, which is found to 0.4 decrease to negligible values in less than a second. The 5 0.2 transient behavior of the magnitude estimate 811 is shown a 3 in Fig. 4, where the solid line is the parameter response, g o and the dashed line is the response predicted from a simm ulation of an approximate system. The approximate sys-0.2 tem is composed of the control law (7), ( S ) , and the nonlinear approximation (9), (10). The frequency estimate -0.r 812 is shown in Fig. 5. The matches between the actual -0.f and approximate behaviors is good, and the degree of match is similar to those for other adaptive systems [SI. -0l The magnitude estimate for the third harmonic 831 is 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (seconds) shown in Fig. 6 and the phase estimate 832 in Fig. 7. Figure 3: Plant output (y) Note that the magnitude e;, was set to -1, and the estiThe performance of the algorithm is examined via simumate of the magnitude converged to 1, with the estimate lation. We consider a situation in which P ( s ) = lOO/(s+ of the phase converging to - T , or -180'. Again, the approximations are very good, and although the nonlinl o o ) , W I = 100, d l = 1, and d3 = -1. The parameters 2618 gl
I
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ear effects are significant, the design based on the linear approximation is adequate to obtain convergence of the system
-
t
-0.06
-0.08 0
0.5
I I
1.5
Time (seconds)
Figure 8: Microphone signal -0.41
0
"
0.1
"
"
0.2 0.3 0.4 0.5 0.6 Time (seconds)
"
0.7
0.8
'
0.9
I
1
Figure 6: Magnitude of the third harmonic (Ow 1
1 0-5 .-
I 0
0.5
1
1.5
Time (seconds)
Figure 9: Magnitudse of the fundamental 0.8
0.9
Figure 7: Phase of the third harmonic
(032)
0
0.1
0.5 0.6 Time (seconds)
0.2 0.3 0.4
0.7
(011)
1
were obtained at 16 different frequencies, spaced logarithmically between 32.5 Hz a.nd 1kHz. In other experiments, 5. Experimental Results the frequency was calculated from a 50-tap finite - response _ impulse response model obtained with an adaptive idenThe scheme was implemented on an experimental actification algorithm and at white noise input. This identitive noise control system developed a t the University of fication procedure took longer. The matrices G1 and G3 Utah. The algorithm was coded in assembly language on were adjusted in real-time, based on the frequency estia Motorola DSP96002 32-bit floating-point digital signal mate 012. Values from a look-up table were interpolated processorl The sampling rate was set at 8 kHz. A sinlinearly as needed. UPdatse ofthe matrices was Performed gle bookshelf speaker with a 4-inch low-frequency driver, every 8 samples. Because of the digital implementation, located approximately 2 ft away from the error microa discrete-time equivalent of the algorithm was implephone, generated a periodic signal constituting the noise mented, with the z-domain poles placed in the vicinity source. The microphone signal was passed through an of z = 0.995 for the para.meters of the fundamental and anti-aliasing filter and sampled by a self-calibrating 16z = 0.99 for the parameters of the third harmonic. bit analog-to-digital converter before being sent to the The results of one experiment are shown on Fig. 8, in DSP system. The controller output signal was sent to a which the signal at the error microphone is plotted as a noise cancelling speaker placed approximately 1 ft away function of time. The algorithm is not engaged until 0.5 from the microphone. Only a single error sensing microsec. in the experiment, so that the amount of noise before phone signal was used. compensation can be judged. The frequency of the fundaThe frequency response of the plant was determined mental is 110 Hz. The plot shows that the algorithm, once during a rapid calibration phase in which pure sinusoidal engaged, cancels the noise within a fraction of a second. tones were applied to the noise cancellation speaker, and The adaptive parameters for the fundamental are shown the responses were measured by the microphone signal. in Fig. 9 (magnitude estimate 811) and Fig. 10 (frequency The real and imaginary parts of the frequency response 2619
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0.12 I
0
I
0.5
1
I._
I .5
0
Time (seconds)
0.05
3
I
-
,
Figure 12: Phase of the third harmonic
I
-0.05E =
(U
s.-
-0.1
-
-0.15
-
7. References
C
5
(032)
nonlinear time-varying system. However, we showed that an approximate nonlinear time-invariant system provided a very accurate representation of the dynamic behavior of the system. A further approximation of the system through linearization was useful for the selection of the design parameters. The algorithm was tested experimentally on an active noise control system at the University of Utah to demonstrate the success of the method in a practical environment.
1
0
W
1.5
Time (seconds)
Figure 10: Frequency of the fundamental (012) 0.1
1
0.5
S.M. Kuo and D.R. Morgan, Active Noise Control Systems: Algorilhms and DSP Implementations, New York, Wiley, 1996.
”.*
0
0.5
1
1.5
Time (seconds)
K.S. Narendra and A. Annaswamy, Stable Adaptive
Figure 11: Magnitude of the third harmonic (03,)
Systems, Prentice-Hall, Englewood Cliffs, NJ, 1989.
estimate 012). The units of the frequency estimate are given in radians/sample; L e , 27r . 110/8000 = 0.086. The initial frequency estimate corresponded to 115 He. Other parameters were initially set to zero. Shown in Fig. 11 and 12 are the parameters related to the third harmonic (magnitude estimate 031 and phase estimate 032 (in rad)). It was found that the frequency of the disturbance could vary over a wide range, once the algorithm had locked onto the disturbance frequency. Techniques from phaselocked loops could be used to expand the lock-in range [7] but were not implemented. Other experiments employing several harmonics as well as frequency estimation using high-order harmonics provided results comparable to those shown.
G. Feng and M. Palaniswamy, “Unified Treatment of Internal Model Principle Based Adaptive Control Algorithm,” Int. J. Control, vol. 54, no. 4, pp. 883-901, 1991. G. Feng and M . Palaniswamy, “A Stable Adaptive Implementation of the Internal Model Principle,” IEEE Trans. on Automatic Control, vol. 37, no. 8, pp. 12201225, 1992. G.B.B. Chaplin and R.A. Smith, “Method of and Apparatus for Cancelling Vibrations from a Source of Repetitive Vibrations,” U. S. Patent 4,566,118, Jan. 21, 1986. M. Bodson and S.C. Douglas, “Adaptive Algorithms for the Rejection of Periodic Disturbances with Un-
6. Conclusions
known Frequency,” Automatica, vol. 33, no. 12, pp. 2213-2221, 1997.
An adaptive algorithm is proposed for the rejection of periodic disturbances of unknown frequency. For simplicity, the algorithm is described for a noise consisting of a fundamental component and a third harmonic, although the method is easily extended to noises with an arbitrary number of harmonics. As in other solutions to this control problem, the closed-loop system is a complex
J.R. Smith, Modern Communzcations Circuits, Mc Graw-Hill, New York, NY, 1997.
S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness, Prentice-Hall, Englewood Cliffs, NJ, 1989.
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